Acyclic Systems of Permutations and Fine Mixed Subdivisions of Simplices

Abstract

A fine mixed subdivision of a (d - 1)-simplex T of size n gives rise to a system of (2d) permutations of [n] on the edges of T, and to a collection of n unit (d - 1)-simplices inside T. Which systems of permutations and which collections of simplices arise in this way? The Spread Out Simplices Conjecture of Ardila and Billey proposes an answer to the second question. We propose and give evidence for an answer to the first question, the Acyclic System Conjecture. We prove that the system of permutations of T determines the collection of simplices of T. This establishes the Acyclic System Conjecture as a first step towards proving the Spread Out Simplices Conjecture. We use this approach to prove both conjectures for n=3 in arbitrary dimension. © 2013 Springer Science+Business Media New York.